The broad objective of the proposed research is to incorporate non- linearity in some recently developed linear mathematical models for mechanical ventilation with the goal of developing models that correlates better with clinical data. In particular, using non-linear relationship between airflow and resistive pressures, new models for two- compartment pressure preset ventilation and non-passive, one compartment mechanical ventilation, will be formulated and solved. These models will then be used to develop mathematical models for tracheal gas insuffation (TG) that permits the practitioner better understand the relationship between the clinically set inputs, such as ventilator, ventillator waveform, inspiratory time, etc. and key clinical outputs such as minute ventilation, mean alveolar pressure, autoPEEP, bloodgases, etc.